Answer
$P(X|Y)=\frac{.6\times.6}{.6\times.6+.3\times .4}= .75$
Work Step by Step
Accordint to Bayes' theorem:
$P(A|B)=\frac{P(B|A)P(A)}{P(B|A)P(A)+P(B|A')P(A')}$
Here, we have
$P(X|Y)= .6$
$P(Y')= .4$
$P(X|Y')= .3$
Be aware, that X and Y equal to B and A in the definition of the theorem, respectively.
We know, that $P(Y)=1-P(Y')=1-.4=.6$
We can substitute into the definition:
$P(X|Y)=\frac{.6\times.6}{.6\times.6+.3\times .4}= .75$
